= Solution
{c}
For a <Conformal Killing vector field>, the same <metric compatibility> calculation gives
$$
\mathcal L_Xg_{\mu\nu}=\nabla_\mu X_\nu+\nabla_\nu X_\mu=2\lambda g_{\mu\nu}.
$$
Thus its <Conformal Killing equation> is $\nabla_{(\mu}X_{\nu)}=\lambda g_{\mu\nu}$. Taking the <tensor contraction> with the <inverse metric>, and using $g^{\mu\nu}g_{\mu\nu}=4$, determines the <conformal Killing factor>:
$$
\boxed{\nabla_{(\mu}X_{\nu)}=\lambda g_{\mu\nu},\qquad \lambda=\frac14\nabla_\mu X^\mu.}
$$
Parentheses include the factor $1/2$ in <tensor index symmetrization>.
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